Consider the following relations: R = (x, y) | x, y are real numbers and x = wy for some rational number w ; S = ( mn, pq ) | m, n, p and q are integers such that n, q 0 and qm = pn . Then

JEE Main 2010 Mathematics question

Consider the following relations: are real numbers and for some rational number w ; S = \left\{\left(\frac mn, \frac pq\right) | m, n, p and q are integers such that n, q \neq 0 and qm = pn\right\}. Then

(1)neither R nor S is an equivalence relation
(2) is an equivalence relation but is not an equivalence relationCorrect
(3) and both are equivalence relations
(4)R is an equivalence relation but S is not an equivalence relation

Answer: (2)

Solution

R is not symmetric: (0, 1) ∈ R since 0 = 0·1, but (1, 0) ∉ R since 1 = w·0 is impossible — so R is not an equivalence relation. S relates two fractions exactly when they are equal ($ mn =

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Question ID VK-039274